Ambient Metrics Research Articles

Min-max theory for capillary surfaces

Abstract We develop a min-max theory for the construction of capillary surfaces in 3-manifolds with smooth boundary. In particular, for a generic set of ambient metrics, we prove the existence of nontrivial, smooth, almost properly embedded surfaces with any given constant mean curvature 𝑐, and with smooth boundary contacting at any given constant angle 𝜃. Moreover, if 𝑐 is nonzero and 𝜃 is not π 2 \frac{\pi}{2} , then our min-max solution always has multiplicity one. We also establish a stable Bernstein theorem for minimal hypersurfaces with certain contact angles in higher dimensions.

Variational characterizations of invariant submanifolds in Sasaki manifolds

In this note, we generalize our previous results in [4] and [5] from Kähler manifold to Sasaki manifold. We prove that for a submanifold tangent to the Reeb vector field in a Sasaki manifold, the area functional, as a functional of the ambient metrics, is stationary if and only if it is an invariant submanifold. We also show that if the submanifold has dimension 3, then it is basic stable if and only if it is an invariant submanifold.

The Weighted Ambient Metric

We prove the existence and uniqueness of weighted ambient metrics and weighted Poincaré metrics for smooth metric measure spaces.

Habitat associations and impacts on a juvenile fish host by a temperate gnathiid isopod.

The distribution and abundance of organisms is typically shaped by multiple biotic and abiotic processes. Micropredators are parasite-like organisms that are smaller than their hosts and/or prey and feed on multiple hosts during a given life stage. Unlike typical parasites, however, they spend much or most of their time free-living, associating only temporarily with hosts. In the ocean, micropredators can impact multiple fish species, and in particular can have significant lethal and sub-lethal effects on newly settled fish. Although gnathiid isopods are abundant and primary micropredators in coral reef ecosystems, their impacts are relatively unexplored within sub-tidal temperate rocky reefs. We investigated the distribution of juvenile gnathiid isopods along sub-tidal temperate rocky reefs and tested trap methodology. We also quantified both the sub-lethal and lethal impacts of feeding-stage juvenile gnathiid isopods on juvenile, post-settlement reef fish, Heterostichus rostratus (giant kelpfish). We were most interested in determining the relationship between gnathiid infestation level and fish swimming performance, in particular swimming metrics relevant to predator avoidance maneuvers. We found that Gnathia tridens was present in rocky reefs rather than embayments along the Southern California coastline and that within rocky reefs, gnathiids occurred in the highest densities in lighted traps. Surprisingly, we observed almost no influence of fish size or gnathiid sub-lethal infestation level on ambient or burst swimming performance metrics. However, burst duration was reduced by gnathiid infestation, which is important in predator avoidance. There were significant differences in survivorship among small fish compared to large fish as a result of gnathiid infestation. Larger fish survived higher numbers of gnathiids than smaller fish, indicating that parasite-induced mortality is greater for smaller fish. Investigations of the effects of micropredators on subsequent predator-mediated mortality, including the susceptibility of fishes and their individual responses to micropredators, can further contribute to our understanding of processes affecting recruitment in resident reef fish populations. Further research, especially within temperate sub-tidal ecosystems, is needed to understand and highlight the overlooked importance of micropredation in shaping fish populations within a reefscape.

APExpose_DE, an air quality exposure dataset for Germany 2010\u20132019

Exposure to poor air quality is considered a major influence on the occurrence of cardiovascular and respiratory diseases. Air pollution has also been linked to the severity of the effects of epidemics such as COVID-19 caused by the SARS-CoV-2 virus. Epidemiological studies require datasets of the long-term exposure to air pollution. We present the APExpose_DE dataset, a long-term (2010–2019) dataset providing ambient air pollution metrics at yearly time resolution for NO2, NO, O3, PM10 and PM2.5 at the NUTS-3 spatial resolution level for Germany (corresponding to the Landkreis or Kreisfreie Stadt in Germany, 402 in total).

Explicit ambient metrics and holonomy

We present three large classes of examples of conformal structures for which the equations for the Fefferman-Graham ambient metric to be Ricci-flat are linear PDEs, which we solve explicitly. These explicit solutions enable us to discuss the holonomy of the corresponding ambient metrics. Our examples include conformal pp-waves and, more importantly, conformal structures that are defined by generic rank 2 and 3 distributions in respective dimensions 5 and 6. The corresponding explicit Fefferman-Graham ambient metrics provide a large class of metrics with holonomy equal to the exceptional non-compact Lie group $\mathbf{G}_2$ as well as ambient metrics with holonomy contained in $\mathbf{Spin}(4,3)$.

Mean curvature flows in manifolds of special holonomy

We study the uniqueness of minimal submanifolds and the stability of the mean curvature flow in several well-known model spaces of manifolds of special holonomy. These include the Stenzel metric on the cotangent bundle of spheres, the Calabi metric on the cotangent bundle of complex projective spaces, and the Bryant--Salamon metrics on vector bundles over certain Einstein manifolds. In particular, we show that the zero sections, as calibrated submanifolds with respect to their respective ambient metrics, are unique among compact minimal submanifolds and are dynamically stable under the mean curvature flow. The proof relies on intricate interconnections of the Ricci flatness of the ambient space and the extrinsic geometry of the calibrated submanifolds.

Fefferman–Graham ambient metrics of Patterson–Walker metrics

Given an $n$-dimensional manifold $N$ with an affine connection $D$, we show that the associated Patterson-Walker metric $g$ on $T^*N$ admits a global and explicit Fefferman-Graham ambient metric. This provides a new and large class of conformal structures which are generically not conformally Einstein but for which the ambient metric exists to all orders and can be realized in a natural and explicit way. In particular, it follows that Patterson-Walker metrics have vanishing Fefferman-Graham obstruction tensors. As an application of the concrete ambient metric realization we show in addition that Patterson-Walker metrics have vanishing Q-curvature.

Cartan\u2019s incomplete classification and an explicit ambient metric of holonomy mathrm{G}_2^*

In his 1910 “Five Variables” paper, Cartan solved the equivalence problem for the geometry of (2, 3, 5) distributions and in doing so demonstrated an intimate link between this geometry and the exceptional simple Lie groups of type mathrm{G}_2. He claimed to produce a local classification of all such (complex) distributions which have infinitesimal symmetry algebra of dimension at least 6 (and which satisfy a natural uniformity condition), but in 2013 Doubrov and Govorov showed that this classification misses a particular distribution mathbf {E}. We produce a closed form for the Fefferman–Graham ambient metric {smash {{smash {widetilde{g}}}}}_{mathbf {E}} of the conformal class induced by (a real form of) mathbf {E}, expanding the small catalogue of known explicit, closed-form ambient metrics. We show that the holonomy group of smash {{smash {widetilde{g}}}_{mathbf {E}}} is the exceptional group {smash {mathrm{G}}_2^*} and use that metric to give explicitly a projective structure with normal projective holonomy equal to that group. We also present some simple but apparently novel observations about ambient metrics of general left-invariant conformal structures that were used in the determination of the explicit formula for smash {{smash {widetilde{g}}}_{mathbf {E}}}.

Highly symmetric 2-plane fields on 5-manifolds and 5-dimensional Heisenberg group holonomy

Nurowski showed that any generic 2-plane field D on a 5-manifold M determines a natural conformal structure cD on M; these conformal structures are exactly those (on oriented M) whose normal conformal holonomy is contained in the (split, real) simple Lie group G2. Graham and Willse showed that for real-analytic D the same holds for the holonomy of the real-analytic Fefferman–Graham ambient metric of cD, and that both holonomy groups are equal to G2 for almost all D. We investigate here independently interesting 2-plane fields for which the associated holonomy groups are a proper subgroup of G2.Cartan solved the local equivalence problem for 2-plane fields D and constructed the fundamental curvature tensor A for these objects. He furthermore claimed to describe locally all D whose infinitesimal symmetry algebra has rank at least 6 and gave a local quasi-normal form, depending on a single function of one variable, for those that furthermore satisfy a natural degeneracy condition on A, but Doubrov and Govorov recently rediscovered a counterexample to Cartanʼs claim. We show that for all D given by Cartanʼs alleged quasi-normal form, the conformal structures cD induced via Nurowskiʼs construction are almost Einstein, that we can write their ambient metrics explicitly, and that the holonomy groups associated to cD are always the 5-dimensional Heisenberg group, which here acts indecomposably but not irreducibly. (Not all of these properties hold, however, for Doubrov and Govorovʼs counterexample.) We also show that the similar results hold for the related class of 2-plane fields defined on suitable jet spaces by ordinary differential equations z′(x)=F(y″(x)) satisfying a simple genericity condition.

Parallel tractor extension and ambient metrics of holonomy split $G_2$

The holonomy of the ambient metrics of Nurowski’s conformal structures associated to generic real-analytic 2-plane fields on oriented 5-manifolds is investigated. It is shown that the holonomy is always contained in the split real form $G_2$ of the exceptional Lie group, and is equal to $G_2$ for an open dense set of 2-plane fields given by explicit conditions. In particular, this gives an infinite-dimensional family of metrics of holonomy equal to split $G_2$. These results generalize work of Leistner-Nurowski. The inclusion of the holonomy in $G_2$ is established by proving an ambient extension theorem for parallel tractors in the context of conformal geometry in general signature and dimension, which is expected to be of independent interest.

Ambient metrics with exceptional holonomy

We present conformal structures in signature (3, 2) for which the holonomy of the Fefferman-Graham ambient metric is equal to the non-compact exceptional Lie group G2(2). We write down the resulting 8-parameter family of G2(2)-metrics in dimension seven explicitly in an appropriately chosen coordinate system on the ambient space.

Regression Calibration for Classical Exposure Measurement Error in Environmental Epidemiology Studies Using Multiple Local Surrogate Exposures

Environmental epidemiologic studies are often hierarchical in nature if they estimate individuals' personal exposures using ambient metrics. Local samples are indirect surrogate measures of true local pollutant concentrations which estimate true personal exposures. These ambient metrics include classical-type nondifferential measurement error. The authors simulated subjects' true exposures and their corresponding surrogate exposures as the mean of local samples and assessed the amount of bias attributable to classical and Berkson measurement error on odds ratios, assuming that the logit of risk depends on true individual-level exposure. The authors calibrated surrogate exposures using scalar transformation functions based on observed within- and between-locality variances and compared regression-calibrated results with naive results using surrogate exposures. The authors further assessed the performance of regression calibration in the presence of Berkson-type error. Following calibration, bias due to classical-type measurement error, resulting in as much as 50% attenuation in naive regression estimates, was eliminated. Berkson-type error appeared to attenuate logistic regression results less than 1%. This regression calibration method reduces effects of classical measurement error that are typical of epidemiologic studies using multiple local surrogate exposures as indirect surrogate exposures for unobserved individual exposures. Berkson-type error did not alter the performance of regression calibration. This regression calibration method does not require a supplemental validation study to compute an attenuation factor.

A SUB-PRODUCT CONSTRUCTION OF POINCARÉ–EINSTEIN METRICS

Given any two Einstein (pseudo-)metrics, with scalar curvatures suitably related, we give an explicit construction of a Poincaré–Einstein (pseudo-)metric with conformal infinity the conformal class of the product of the initial metrics. We show that these metrics are equivalent to ambient metrics for the given conformal structure. The ambient metrics have holonomy that agrees with the conformal holonomy. In the generic case the ambient metric arises directly as a product of the metric cones over the original Einstein spaces. In general the conformal infinity of the Poincaré metric we construct is not Einstein, and so this describes a class of non-conformally Einstein metrics for which the (Fefferman–Graham) obstruction tensor vanishes.

Effect of Leaf Litter on Phosphorus Retention and Hydrological Properties at a First Order Stream in Northeast Oklahoma, USA

ABSTRACT We examined the relationship between phosphorus (P) uptake length (Sw) measured by stable PO4 additions and the level of PO4 enrichment from the additions during summer and fall in a first order stream in the southwestern portion of the Ozark Plateaus in northeast Oklahoma, USA. The y-intercept of this relation was used to more closely approximate ambient Sw and other metrics of P retention efficiency during summer and fall and determine the effect of leaf litter input on P retention efficiency and hydrological properties. Under similar discharge, the presence of the leaf litter during fall decreased water velocity and the dispersion coefficient, while the transient storage area increased compared to summer. An increase in Sw was generally observed with an increase in the level of P enrichment, and the linear relation (y-intercept) estimated Sw at 117 and 86 m during summer and fall. The slope of the linear relation was variable because a 5 μg L−1 increase in P enrichment resulted in a 27 and 5 m increase in Sw estimation during summer and fall. Overall, the input of leaf litter into the stream had a relatively small effect on Sw, but our results suggest P retention was slightly greater during fall compared to summer.

Standard Tractors and the Conformal Ambient Metric Construction

In this paper we relate the Fefferman–Graham ambientmetric construction for conformal manifolds to the approach toconformal geometry via the canonical Cartan connection. We show thatfrom any ambient metric that satisfies a weakening of the usualnormalisation condition, one can construct the conformal standardtractor bundle and the normal standard tractor connection, which areequivalent to the Cartan bundle and the Cartan connection. This resultis applied to obtain a procedure to get tractor formulae for allconformal invariants that can be obtained from the ambient metricconstruction. We also get information on ambient metrics whichare Ricci flat to higher order than guaranteed by the results ofFefferman–Graham.